 
Summary: Research Statement
Ellie Grano
1 Introduction
My research is in topology and knot theory, in particular planar algebras, a new field in
quantum topology. Planar algebras were formally defined by Jones [Jon99]. Kuperberg
independently defined a spider, a similar idea to Jones' planar algebra [Ku96]. My research
is focused on taking a skein theoretic, combinatorial approach to developing new planar
algebras. This follows the Kuperberg Program:
Give a presentation for every interesting planar algebra, and prove as much as possible
about the planar algebra using only its presentation.
2 Background
The most fundamental planar algebra is the TemperleyLieb (T L) planar algebra. Before
the planar algebra came the algebra, which arose from the study of Statistical Mechanics
by Temperley and Lieb [TL71]. The nth TemperleyLieb algebra (T Ln) as a vectorspace
over C has as a basis the diagrams with n noncrossing strands. Multiplication is defined
by vertical stacking and replacing any closed loops by factors of a fixed number C.
The T L planar algebra is the assembly of these T Ln with infinitely many operations
corresponding to the all the ways to connect these diagrams together. These operations are
called "planar tangles". In general, planar algebras may contain more types of diagrams
along with skein relations. Skein relations are relationships between diagrams that are
